Losing-Trick Count

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In the card game contract bridge, the Losing-Trick Count (LTC) is a method of hand evaluation used in situations where a trump suit has been established and shape and fit are more significant than high card points (HCP) in determining the optimum level of the contract. Based on the bidding and a set of empirical rules, the number of "losing tricks" held in each of the partnership's hands is estimated and their sum deducted from 24; the result is the number of tricks the partnership can expect to take when playing in their established suit, assuming normal breaks and assuming required finesses work about half the time.[1]

Origins[edit]

The origins of the Losing Trick Count—without that name—can be traced back at least to 1910 in Joseph Bowne Elwell's book Elwell on Auction Bridge. In the preface (page v), Elwell mentions chapters on "Estimating the Values of Hands". The sections later in the book (pages 80–89) are mostly tables with differing titles beginning with the word "Estimating" but ending differently. Elwell sets out a scheme for counting losers in trump contracts that looks very much like the simple basic counting method given below.

The term "Losing Trick Count" was originally put forward by the American F. Dudley Courtenay in his 1934 book The System the Experts Play (which ran to at least 18 printings). On page two among various Acknowledgments, the author writes: 'To Mr. Arnold Fraser-Campbell the author is particularly indebted for permission to use material and quotations from his manuscript in which is described his method of hand valuation by counting losing tricks, and from which the author has developed the Losing Trick Count described herein.' From this we may speculate that Elwell's ideas filtered through Fraser-Campbell to Courtenay.

The Englishman George Gordon Joseph Walshe contacted Courtenay about issuing a British edition. Together they edited the American edition and retitled it The Losing Trick Count for the British market. This title went through dozens of printings and remained in print for two decades. (Subsequently it has been republished by print-on-demand re-publishers.)

LTC was popularised by Maurice Harrison-Gray in Country Life magazine in the 1950s and 1960s. In recent decades, others have suggested refinements to the basic counting method.

Basic counting method[edit]

The estimated number of losing tricks (LTC) in one's hand is determined by examining each suit and assuming that an ace will never be a loser, nor will a king in a 2+ card suit, nor a queen in a 3+ card suit; accordingly

  • a void = 0 losing tricks.
  • a singleton other than an A = 1 losing trick.
  • a doubleton AK = 0; Ax or Kx = 1; xx = 2 losing tricks.
  • a three card suit AKQ = 0; AKx, AQx or KQx = 1 losing trick.
  • a three card suit Axx, Kxx or Qxx = 2; xxx = 3 losing tricks.

(Some authorities treat Qxx as 3 losers unless the Q is "balanced" by an A in another suit.) LTC also assumes that no suit can have more than 3 losing tricks and so suits longer than three cards are judged according to their three highest cards. It follows that hands without an A, K or Q have a maximum of 12 losers but may have fewer depending on shape, e.g.  J x x x  J x x  J x x  J x x has 12 losers (3 in each suit), whereas  x x x x x  —  x x x x  x x x x has only 9 losers (3 in all suits except the void which counts no losers).

Until further information is derived from the bidding, assume that a typical opening hand by partner contains 7 losers, e.g.  A K x x x  A x x x  Q x  x x, has 7 losers (1 + 2 + 2 + 2 = 7).

Example[edit]

To determine how high to bid, responder adds the number of losers in his hand to the assumed number in opener's hand (7); the total number of losers arrived at by this sum is subtracted from 24 and the result is estimated to be the total number of tricks available to the partnership.

Thus following an opening bid of 1:

  • partner jumps to game in 4 with no more than 7 losers in his own hand and a fit with partner's heart suit (7 + 7 = 14 subtract from 24 = 10 tricks).
  • With 8 losers in hand and a fit, responder bids 3 (8 + 7 = 15 subtract from 24 = 9 tricks).
  • With 9 losers and a fit, responder bids 2 (9 + 7 = 16 subtract from 24 = 8 tricks).
  • With only 5 losers and a fit (5 + 7 = 12 subtract from 24 = 12 tricks), a slam is possible so responder may bid straight to 6 if preemptive bidding seems appropriate or take a slower forcing approach.

Refinements[edit]

Thinking that the method tended to overvalue unsupported queens and undervalue supported jacks, Eric Crowhurst and Andrew Kambites refined the scale, as have others:

  • AQ doubleton = ½ loser according to Ron Klinger.
  • KQ doubleton = 1 loser (obvious).
  • Kx doubleton = 1½ losers according to others.
  • AJ10 = 1 loser according to Harrison-Gray.
  • KJ10 = 1½ losers according to Bernard Magee.
  • Qxx = 3 losers (or possibly 2.5) unless trumps.
  • Subtract a loser if there is a known 9-card trump fit.

In his book The Modern Losing Trick Count, Ron Klinger advocates adjusting the number of loser based on the control count of the hand believing that the basic method undervalues an ace but overvalues a queen and undervalues short honor combinations such as Qx or a singleton king. Also it places no value on cards jack or lower.

New Losing-Trick Count (NLTC)[edit]

A "New" Losing-Trick Count (NLTC) was introduced in The Bridge World, May 2003, by Johannes Koelman. Designed to be more precise than LTC, the NLTC method of hand evaluation utilizes the concept of "half-losers", and it distinguishes between 'Ace-losers', 'King-losers' and 'Queen-losers.' NLTC intrinsically assigns greater value to Aces than it assigns to Kings, and it assigns greater value to Kings than it assigns to Queens. Some users of LTC make adjustments to the loser count to compensate for the imbalance of Aces and Queens held. Koelman argues that adjusting a hand's value for the imbalance between Aces and Queens held isn't the same as correcting for the imbalance between Aces and Queens missing. Because of singletons and doubletons, missing Aces that add losers tend to outnumber missing Queens that add losers.[2]


NLTC differs from LTC in two significant ways. First, NLTC uses a different method to count losers (explanation and loser-count lists below). Consequently with NLTC, the number of losers in a singleton or doubleton suit can exceed the number of cards in the suit. Second, with NLTC the number of combined losers between two hands is subtracted from 25, not from 24 (explanation below), to predict the number of tricks the two hands will produce when declarer plays the hand in the agreed trump suit. As with LTC, the NLTC formula assumes normal suit breaks, it assumes that required finesses work about half the time, and it must only be applied after an 8-card trump fit or better is discovered. When counting NLTC losers in a hand, consider only the three highest ranking cards in each suit:

  • Count 1.5 losers for a missing A in a suit of at least 1 card in length
  • Count 1.0 losers for a missing K in a suit of at least 2 cards in length
  • Count 0.5 losers for a missing Q in a suit of at least 3 cards in length
  • Count 0 losers for a void suit


The following three basic hands, which are all valued equally with LTC, are often referred to when illustrating the differences between the A,K,Q values with LTC and NLTC:

♠Axxx ♥Axx ♦Axx ♣Axx - 8 LTC losers, but only 6 NLTC losers
♠Kxxx ♥Kxx ♦Kxx ♣Kxx - 8 LTC losers, and also 8 NLTC losers
♠Qxxx ♥Qxx ♦Qxx ♣Qxx - 8 LTC losers, but 10 NLTC losers


Here is the basic NLTC list. For simplicity, cards below the rank of Queen are represented by "x":

Void - 0 losers
AKQ - 0 losers
AKx - 0.5 losers (missing Q)
AQx - 1.0 losers (missing K)
Axx - 1.5 losers (missing K,Q)
KQx - 1.5 losers (missing A)
Kxx - 2.0 losers (missing A,Q)
Qxx - 2.5 losers (missing A,K)
xxx - 3.0 losers (missing A,K,Q)
AK doubleton - 0 losers
AQ doubleton - 1.0 losers (missing K)
Ax doubleton - 1.0 losers (missing K)
KQ doubleton - 1.5 losers (missing A)
Kx doubleton - 1.5 losers (missing A)
Qx doubleton - 2.5 losers (missing A,K)
xx doubleton - 2.5 losers (missing A,K)
A singleton - 0 losers
K singleton - 1.5 losers (missing A)
Q singleton - 1.5 losers (missing A)
x singleton - 1.5 losers (missing A)


All singletons, except singleton A, are initially counted as 1.5 losers, and all doubletons that are missing both the A and K are initially counted as 2.5 losers. Professional bridge player, Kevin Wilson, explains this concept of a suit that contains more losers than it contains cards: "Think about how much of declarer play is about timing. When you're missing an Ace, you're losing more than just a trick; you're losing timing because the King, Queen and Jack that you might hold can't score immediate tricks. First you must force out the Ace [and when the opponents win their Ace, they might immediately score more tricks, or they might establish winning tricks for later in the play]. The idea of 1.5 losers for a singleton [and 2.5 losers for a doubleton] should be within your grasp!"[3] In Kevin's article, he coins the term "modified" losing-trick count, or MLTC.


The following is a more extensive and detailed NLTC list. This list is essentially the same as the basic list, but it displays J's and T's which technically don't affect the losing-trick count. This list also displays an "(x)" after each of the three-card listings, which reminds users that the loser count for three-card suits and suits of three-plus length is the same:

Void - 0 losers
AKQ(x) - 0 losers
AKJ(x) - 0.5 losers
AKT(x) - 0.5 losers
AKx(x) - 0.5 losers
AQJ(x) - 1.0 losers
AQT(x) - 1.0 losers
AQx(x) - 1.0 losers
AJT(x) - 1.5 losers
AJx(x) - 1.5 losers
ATx(x) - 1.5 losers
Axx(x) - 1.5 losers
KQJ(x) - 1.5 losers
KQT(x) - 1.5 losers
KQx(x) - 1.5 losers
KJT(x) - 2.0 losers
KJx(x) - 2.0 losers
KTx(x) - 2.0 losers
Kxx(x) - 2.0 losers
QJT(x) - 2.5 losers
QJx(x) - 2.5 losers
QTx(x) - 2.5 losers
Qxx(x) - 2.5 losers
JTx(x) - 3.0 losers
Jxx(x) - 3.0 losers
Txx(x) - 3.0 losers
xxx(x) - 3.0 losers
AK doubleton - 0 losers
AQ doubleton - 1.0 losers
AJ doubleton - 1.0 losers
AT doubleton - 1.0 losers
Ax doubleton - 1.0 losers
KQ doubleton - 1.5 losers
KJ doubleton - 1.5 losers
KT doubleton - 1.5 losers
Kx doubleton - 1.5 losers
QJ doubleton - 2.5 losers
QT doubleton - 2.5 losers
Qx doubleton - 2.5 losers
JT doubleton - 2.5 losers
Jx doubleton - 2.5 losers
Tx doubleton - 2.5 losers
xx doubleton - 2.5 losers
A singleton - 0 losers
K singleton - 1.5 losers
Q singleton - 1.5 losers
J singleton - 1.5 losers
T singleton - 1.5 losers
x singleton - 1.5 losers


As with LTC, players seeking greater accuracy can also make adjustments with NLTC. LTC normally uses a one-loser "resolution" (i.e. degree of accuracy), and players who adjust with LTC commonly adjust in ½-loser increments. To compare, NLTC normally uses a ½-loser resolution, and typical NLTC adjustments are made in ¼-loser increments. For more accuracy players can also adjust using smaller increments. J's and T's are initially assigned no value in losing-trick counts, but these lower honors can be valuable (not to mention the value of intermediates like 9's, 8's and 7's). J's and T's are more valuable when they're together in the same suit, and they're most valuable when they support higher honors in the suit. Other holdings that possess no initial losing-trick count value, but can be considered for upgrades, include unprotected K's, Q's and J's in short suits. As with other methods of hand evaluation, players can either upgrade or downgrade the value of a given holding based on the ensuing auction. With no additional information available, the following is a list of plausible initial NLTC adjustments:

AKJ - ¼-loser upgrade
AKT - ⅛-loser upgrade
AQJ - ¼-loser upgrade
AQT - ⅛-loser upgrade
AJT - ¼-loser upgrade
AJx - ⅛-loser upgrade
KQJ - ¼-loser upgrade
KQT - ⅛-loser upgrade
KJT - ¼-loser upgrade
KJx - ⅛-loser upgrade
QJT - ¼-loser upgrade
QJx - ⅛-loser upgrade
JTx - ⅛-loser upgrade
AQ doubleton - ¼-loser upgrade
AJ doubleton - ⅛-loser upgrade
KQ doubleton - ¼-loser upgrade
KJ doubleton - ⅛-loser upgrade
QJ doubleton - ¼-loser upgrade
Qx doubleton - ⅛-loser upgrade
K singleton - ¼-loser upgrade
Q singleton - ⅛-loser upgrade


As previously stated, NLTC uses a value of 25 (instead of 24 with LTC) in the formula for determining the trick-taking potential for two hands. Here's a basic pair of hands that helps illustrate why:
♠ xxxx ♥ xxx ♦ xxx ♣ xxx
♠ xxxx ♥ xxx ♦ xxx ♣ xxx

With both LTC and NLTC, the combined loser count with these two very weak and flat-shaped hands is 24 (12 losers in each hand). According to the LTC formula, there is no trick-taking potential with these hands (24-24 combined losers = 0 winning tricks). We must remember, however, that both forms of the losing-trick count are used only after the partnership knows it has an 8-card fit or better. In addition, losing-trick count predictions assume that all suits will break normally. In this example, given we possess an 8-card spade fit, and assuming the outstanding spades (trumps) split 3-2, the defenders can't prevent the (hypothetical) declarer from scoring one trump trick with these otherwise worthless hands. A losing-trick count formula that doesn't predict one winning trick with these two hands should give a theoretician concern (*). With NLTC we deduct the total combined losers from 25, not from 24, so the NLTC formula accurately predicts the trick-taking potential of these two hands (25-24 losers = 1 winner).

(*) These two example hands are flat shaped, and flat-shaped hands often play better in notrump. Losing trick counts are not inherently designed for notrump hand evaluation.[4] Instead, losing trick counts are intended to be used primarily for suit contract evaluations, particularly when one or both hands are unbalanced. In fact, when one partner has 12 losers - which can occur with 4333 shape, as in the example above - LTC can never predict 13 tricks.[5] NLTC however can predict a grand slam with balanced hands (see additional example hand below). For more information about NLTC, including new losing-trick counts in balanced hands, refer to Lawrence Diamond's Mastering Hand Evaluation.[6]


Also similar to LTC, NLTC users may employ an alternate formula to determine the appropriate contract level for two fitting hands. The NLTC alternate formula is: 19 (instead of 18 with LTC) minus the sum of the losers in the two hands = the projected safe contract level when declarer plays the hand in the agreed trump suit. So, 7.5 losers opposite 7.5 losers leads to: 19-(7.5+7.5) = 19-15 = 4 (4-level contract). Players who use the LTC variation of this formula (i.e. 18 - total combined losers = suggested safe contract level) will recognize the difference between 25 and 19 as the number of tricks required by declarer to secure a "book", which is 6.


The following two lists show typical NLTC guidelines for hands of increasing strength. These guides assume the partnership has an 8-card or longer major-suit fit, and that responder possesses the values to bid. For minor-suit game-invitational and game-forcing actions, the required loser counts should be upgraded by one loser (trick). First consider opener's hand. A hand with opening strength - particularly an unbalanced hand - typically has no more than 7.5 losers:

  • 7.5 losers: minimum values
  • 6.5 losers: game-invitational values
  • 5.5 losers: game-forcing values
  • 4.5 losers: consider investigating slam
  • 3.5 losers: investigate slam

Next consider responder's hand. Opposite partner's opening bid, a supporting hand - particularly an unbalanced hand - typically has no more than 9.5 losers:

  • 9.5 losers: minimum values
  • 8.5 losers: game-invitational values
  • 7.5 losers: game-forcing values
  • 6.5 losers: consider investigating slam
  • 5.5 losers: investigate slam


Here's the full range of combined loser counts and their suggested contract levels, according to NLTC theory:

7.5 losers opposite 10.5 losers: (19-18) = 1-level contract, or (25-18) = 7 tricks
7.5 losers opposite 9.5 losers: (19-17) = 2-level contract, or (25-17) = 8 tricks
7.5 losers opposite 8.5 losers: (19-16) = 3-level contract, or (25-16) = 9 tricks
7.5 losers opposite 7.5 losers: (19-15) = 4-level contract, or (25-15) = 10 tricks
7.5 losers opposite 6.5 losers: (19-14) = 5-level contract, or (25-14) = 11 tricks
7.5 losers opposite 5.5 losers: (19-13) = 6-level contract, or (25-13) = 12 tricks
7.5 losers opposite 4.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
6.5 losers opposite 10.5 losers: (19-17) = 2-level contract, or (25-17) = 8 tricks
6.5 losers opposite 9.5 losers: (19-16) = 3-level contract, or (25-16) = 9 tricks
6.5 losers opposite 8.5 losers: (19-15) = 4-level contract, or (25-15) = 10 tricks
6.5 losers opposite 7.5 losers: (19-14) = 5-level contract, or (25-14) = 11 tricks
6.5 losers opposite 6.5 losers: (19-13) = 6-level contract, or (25-13) = 12 tricks
6.5 losers opposite 5.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
5.5 losers opposite 10.5 losers: (19-16) = 3-level contract, or (25-16) = 9 tricks
5.5 losers opposite 9.5 losers: (19-15) = 4-level contract, or (25-15) = 10 tricks
5.5 losers opposite 8.5 losers: (19-14) = 5-level contract, or (25-14) = 11 tricks
5.5 losers opposite 7.5 losers: (19-13) = 6-level contract, or (25-13) = 12 tricks
5.5 losers opposite 6.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
4.5 losers opposite 10.5 losers: (19-15) = 4-level contract, or (25-15) = 10 tricks
4.5 losers opposite 9.5 losers: (19-14) = 5-level contract, or (25-14) = 11 tricks
4.5 losers opposite 8.5 losers = (19-13) = 6-level contract, or (25-13) = 12 tricks
4.5 losers opposite 7.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
3.5 losers opposite 10.5 losers: (19-14) = 5-level contract, or (25-14) = 11 tricks
3.5 losers opposite 9.5 losers: (19-13) = 6-level contract, or (25-13) = 12 tricks
3.5 losers opposite 8.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
2.5 losers opposite 10.5 losers: (19-13) = 6-level contrast, or (25-13) = 12 tricks
2.5 losers opposite 9.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks
1.5 losers opposite 10.5 losers: (19-12) = 7-level contract, or (25-12) = 13 tricks

Note: as supporting hands become increasingly weaker (i.e. the hand's loser count increases), it becomes increasingly easier, and potentially more effective, to count the hand's "cover cards" instead of counting its losers. For more information about NLTC cover cards, refer to Lawrence Diamond's Mastering Hand Evaluation.[7]


The NLTC solves the problem that the LTC method underestimates the trick taking potential by one on hands with a balance between 'ace-losers' and 'queen-losers'. For instance, the LTC can never predict a grand slam when both hands are 4333 distribution:

KQJ2

W             E

A543
KQ2 A43
KQ2 A43
KQ2 A43

will yield 13 tricks when played in spades on around 95% of occasions (failing only on a 5:0 trump break or on a ruff of the lead from a 7-card suit). However this combination is valued as only 12 tricks using the basic method (24 minus 4 and 8 losers = 12 tricks); whereas using the NLTC it is valued at 13 tricks (25 minus 12/2 and 12/2 losers = 13 tricks). Note, if the west hand happens to hold a small spade instead of the jack, both the LTC as well as the NLTC count would remain unchanged, whilst the chance of making 13 tricks falls to 67%. As a result, NLTC still produces the preferred result.

The NLTC also helps to prevent overstatement on hands which are missing aces. For example:

AQ432

W             E

K8765
KQ 32
KQ52 43
32 KQ54

will yield 10 tricks only, provided defenders cash their three aces. The NLTC predicts this accurately (13/2 + 17/2 = 15 losers, subtracted from 25 = 10 tricks); whereas the basic LTC predicts 12 tricks (5 + 7 = 12 losers, subtracted from 24 = 12).

Second round bids[edit]

Whichever method is being used, the bidding need not stop after the opening bid and the response. Assuming opener bids 1 and partner responds 2; opener will know from this bid that partner has 9 losers (using basic LTC), if opener has 5 losers rather than the systemically assumed 7, then the calculation changes to (5 + 9 = 14 deducted from 24 = 10) and game becomes apparent!

Limitations of the method[edit]

All LTC methods are only valid if trump fit (4-4, 5-3 or better) is evident and, even then, care is required to avoid counting double values in the same suit e.g. KQxx (1 loser in LTC) opposite a singleton x (also 1 loser in LTC).

Regardless which hand evaluation is used (HCP, LTC, NLTC, etc.) without the partners exchanging information about specific suit strengths and suit lengths, a suboptimal evaluation of the trick taking potential of the combined hands will often result. Consider the examples:

QJ53

W             E

AK874
743 A5
KJ2 AQ54
632 54
QJ53

W             E

AK874
743 A5
632 AQ54
KJ2 54

Both layouts are the same, except for the swapping of West's minor suits. So in both cases East and West have exactly the same strength in terms of HCP, LTC, NLTC etc. Yet, the layout on the left may be expected to produce 10 tricks in spades, whilst on a bad day the layout to the right would even fail to produce 9 tricks.

The difference between the two layouts is that on the left the high cards in the minor suits of both hands work in combination, whilst on the right hand side the minor suit honours fail to do so. Obviously on hands like these, it does not suffice to evaluate each hand individually. When inviting for game, both partners need to communicate in which suit they can provide assistance in the form of high cards, and adjust their hand evaluations accordingly. Conventional agreements like helpsuit trials and short suit trials are available for this purpose.

Notes[edit]

  1. ^ Klinger, Ron (2011). The Modern Losing Trick Count (2nd ed.). Sydney, Australia: Modern Bridge Publications. p. 13. ISBN 978-0-9587016-5-5. 
  2. ^ Koelman, Johannes (May 2003). "A New Losing-Trick Count". The Bridge World 74 (8): 28. 
  3. ^ Wilson, Kevin. "Upgrading and Improving Losing Trick Count". BridgeWinners.com. Retrieved 16 February 2014. 
  4. ^ Klinger, Ron (1987). The Modern Losing Trick Count. p. 13. 
  5. ^ Koelman, Johannes (May 2003). "A New Losing-Trick Count". The Bridge World 74 (8): 27. 
  6. ^ Diamond, Lawrence (2015). Mastering Hand Evaluation. Master Point Press. p. 53. 
  7. ^ Diamond, Lawrence (2015). Mastering Hand Evaluation. Master Point Press. 

Further reading[edit]

  • Courtenay, Dudley; Walshe, George (1935). The Losing Trick Count, as used by the leading contract bridge tournament players, with examples of expert bidding and expert play. London: Methuen. p. 176.  Nine editions published between 1935 and 1947. Republished in 2006 as Losing Trick Count - A Book of Bridge Technique by F. Dudley Courtenay, ISBN 978-1-4067-9716-9. Reference: Tim, Bourke; Sugden, John (2010). Bridge Books in English from 1886-2010: an annotated bibliography. Bridge Book Buffs (Cheltenham, England), 711 pages plus supplement. ISBN 978-0-9566576-0-2.  Page 93.
  • Maurice Harrison-Gray articles in Country Life magazine in the 1950s and 1960s.
  • Koelman, Johannes M.V.A. A New Losing-Trick Count. The Bridge World, May 2003, Vol 74, Issue 8, p. 26. 
  • Jones, Jennifer (2011). Losing Trick Count. Jennbridge. 
  • Jones, Jennifer (2012). Losing Trick Count Vol. II. Jennbridge. 
  • Diamond, Lawrence (2015). Mastering Hand Evaluation. Toronto: Master Point Press. ISBN 978-1-77140-153-1.